How about Royal Caribbean or Carnival.
They both offer special things for kids and adults.
I only know RCCL and love this cruiseline, but heard many positive things about Carnival from parents, travelling with their kids.
Hear some other opinions.
Marita

Have you considered Disney? The cabins are roomy, and as your husband doesn't gamble, the fact that the ship doesn't have a casino won't bother him. Yet as Disney is comparatively expensive, I agree that Royal Caribbean (particularly the Voyager class) is a good option.

Carnival is our personal favorite. We tried them all last year. My husband is the mental health anchor for our fox news and he researched all of the kid's programs. Our entire family agreed that Carnival was the best for families with younger kids (our kids ages are 7,4 and 2). Disney is great too, but I found having the Disney environment(always lines for characters) reminded me of the hustle and bustle of being at an amusement park. Plus, in some cases the price is double.